IntRoduction to solid state physics

Practice Course

Sheng Yun Wu

Week 14: Dielectric Properties of Solids

Example 1: Calculate Polarization in a Dielectric Material

This example calculates the polarization P in a dielectric material using the equation:

where χe is the electric susceptibility, E is the applied electric field, and ϵ0 is the permittivity of free space. Polarization describes the dipole moment per unit volume in a dielectric material. This example calculates polarization based on the electric susceptibility and applied electric field, showing how dielectrics respond to external fields.

epsilon_0 = 8.854e-12 # Permittivity of free space (F/m)

def polarization(chi_e, E):

return epsilon_0 * chi_e * E

# Example: Electric susceptibility χ_e = 3, electric field E = 100 V/m

chi_e = 3

E = 100 # V/m

P = polarization(chi_e, E)

print(f"Polarization: {P:.2e} C/m²")

Example 2: Calculate the Dielectric Constant of a Material

This example calculates the dielectric constant ϵr of a material using:

The dielectric constant represents the ratio of a material’s permittivity to the permittivity of free space. This example calculates the dielectric constant using the electric susceptibility, showing how materials increase their ability to store electric energy.

def dielectric_constant(chi_e):

return 1 + chi_e

# Example: Electric susceptibility χ_e = 3

chi_e = 3

epsilon_r = dielectric_constant(chi_e)

print(f"Dielectric Constant: {epsilon_r}")

Example 3: Calculate the Capacitance of a Parallel-Plate Capacitor with a Dielectric

This example calculates the capacitance C of a parallel-plate capacitor with a dielectric using:

where A is the area of the plates, d is the separation between them, and ϵr is the dielectric constant. Capacitance depends on the dielectric constant of the material between the plates. This example calculates the capacitance of a parallel-plate capacitor with a dielectric, illustrating how dielectric materials increase capacitance by storing more charge for the same applied voltage.

def capacitance_parallel_plate(epsilon_r, A, d):

return epsilon_r * epsilon_0 * A / d

# Example: Dielectric constant ε_r = 4, plate area A = 0.01 m², separation d = 1 mm

epsilon_r = 4

A = 0.01 # m²

d = 1e-3 # meters

C = capacitance_parallel_plate(epsilon_r, A, d)

print(f"Capacitance: {C:.2e} F")

Example 4: Calculate the Energy Stored in a Dielectric Capacitor

This example calculates the energy U stored in a dielectric capacitor using:

where C is the capacitance and V is the applied voltage. The energy stored in a dielectric capacitor depends on the capacitance and the applied voltage. This example calculates the energy stored in a capacitor, demonstrating how dielectrics enable capacitors to store more energy.

def energy_stored_in_capacitor(C, V):

return 0.5 * C * V**2

# Example: Capacitance C = 10 μF, applied voltage V = 100 V

C = 10e-6 # F

V = 100 # V

U = energy_stored_in_capacitor(C, V)

print(f"Energy Stored: {U:.2e} J")

Example 5: Calculate the Dipole Moment of a Molecule

This example calculates the dipole moment μ\muμ of a molecule using: μ=qd

where q is the charge and d is the separation between charges.

The dipole moment measures the strength of the separation of charges in a molecule. This example calculates the dipole moment for a molecule with a given charge separation, illustrating the basic concept of molecular dipoles in dielectric materials.

def dipole_moment(q, d):

return q * d

# Example: Charge q = 1.6e-19 C, separation distance d = 1e-10 m

q = 1.6e-19 # C

d = 1e-10 # m

mu = dipole_moment(q, d)

print(f"Dipole Moment: {mu:.2e} C·m")

Example 6: Calculate the Clausius-Mossotti Relation for a Dielectric

This example calculates the Clausius-Mossotti relation, which relates the polarizability α\alphaα of molecules to the dielectric constant:

The Clausius-Mossotti relation connects the macroscopic dielectric constant to the molecular polarizability and density of a dielectric material. This example calculates the Clausius-Mossotti relation for a dielectric material with a given dielectric constant, molecular density, and polarizability.

def clausius_mossotti(epsilon_r, N, alpha):

return (epsilon_r - 1) / (epsilon_r + 2) - (N * alpha) / (3 * epsilon_0)

# Example: Dielectric constant ε_r = 5, molecular density N = 1e28 m⁻³, polarizability α = 1e-40 F·m²

epsilon_r = 5

N = 1e28 # m⁻³

alpha = 1e-40 # F·m²

relation = clausius_mossotti(epsilon_r, N, alpha)

print(f"Clausius-Mossotti Relation: {relation:.2e}")

Example 7: Calculate the Electric Field Inside a Dielectric Sphere

This example calculates the electric field E inside a dielectric sphere placed in an external electric field E0 using:

The electric field inside a dielectric sphere is reduced compared to the external field due to the polarization of the dielectric. This example calculates the internal electric field, showing how the dielectric constant affects field strength within the sphere.

def electric_field_in_sphere(epsilon_r, E_0):

return 3 * E_0 / (epsilon_r + 2)

# Example: Dielectric constant ε_r = 4, external electric field E_0 = 100 V/m

epsilon_r = 4

E_0 = 100 # V/m

E = electric_field_in_sphere(epsilon_r, E_0)

print(f"Electric Field Inside Dielectric Sphere: {E:.2e} V/m")

Example 8: Calculate Piezoelectric Strain in a Material

This example calculates the strain S in a piezoelectric material in response to an applied electric field:

S=d33E

where d33 is the piezoelectric coefficient and E is the applied electric field.

Piezoelectric materials generate strain when an electric field is applied. This example calculates the strain based on the piezoelectric coefficient and applied electric field, showing how piezoelectric materials convert electrical energy into mechanical deformation.

def piezoelectric_strain(d_33, E):

return d_33 * E

# Example: Piezoelectric coefficient d_33 = 1e-12 m/V, applied electric field E = 100 V/m

d_33 = 1e-12 # m/V

E = 100 # V/m

S = piezoelectric_strain(d_33, E)

print(f"Piezoelectric Strain: {S:.2e} m/m")

Example 9: Calculate the Force on a Dielectric in a Capacitor

This example calculates the force F on a dielectric slab inserted into a capacitor using:

When a dielectric is inserted into a capacitor, the electric field exerts a force on it. This example calculates the force on the dielectric based on the dielectric constant, area, applied voltage, and plate separation.

def force_on_dielectric(epsilon_r, A, V, d):

return 0.5 * epsilon_0 * epsilon_r * A * V**2 / d**2

# Example: Dielectric constant ε_r = 4, plate area A = 0.01 m², voltage V = 100 V, separation d = 1 mm

epsilon_r = 4

A = 0.01 # m²

V = 100 # V

d = 1e-3 # m

F = force_on_dielectric(epsilon_r, A, V, d)

print(f"Force on Dielectric: {F:.2e} N")

Example 10: Plot the Electric Displacement Field vs. Electric Field for a Dielectric

This example generates a plot of the electric displacement field D vs. electric field E for a linear dielectric:

The electric displacement field D relates the free charge density to the applied electric field in a dielectric material. This example plots D vs. E, showing the linear relationship in a dielectric material with a constant dielectric constant.

def electric_displacement(epsilon_r, E):

return epsilon_0 * epsilon_r * E

# Example: Dielectric constant ε_r = 5, electric field range from 0 to 100 V/m

E = np.linspace(0, 100, 1000) # V/m

epsilon_r = 5

D = electric_displacement(epsilon_r, E)

plt.plot(E, D)

plt.xlabel("Electric Field (V/m)")

plt.ylabel("Electric Displacement Field (C/m²)")

plt.title("Electric Displacement vs Electric Field")

plt.grid(True)

plt.show()

Summary:

These 10 Python examples cover key topics from Week 14: Dielectric Properties of Solids, including polarization, dielectric constant, capacitance, energy storage, dipole moments, piezoelectricity, the Clausius-Mossotti relation, and forces on dielectrics. These examples help students understand how dielectric materials respond to electric fields, store energy, and generate mechanical responses.

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